thuras et al sound waves 1935

Extraneous Frequencies Generated inAir Carrying Intense Sound Waves * ByA.L.THURAS, R.7,JENKINS and H.7.O'NEIL ligearandconsequently hasmonkesana’combination tonesaregonersted"The prem ofthese extraneous frequencies terns ofthe fundamental rewbres jars, aod decane (rad thesours hosbeen mathematieaty Ectermsed GyRajiegh, Lamb and ethers These eauations have been ppc toanexpaerta hornicamurenenttol thewotharmonicandcombination toneshavebesaRon“Measuresiapeneralsgtwhtheory,butteabesltevalues trelowerthantheealcultedvalues Reet developments inhorntypeloudspeakers! forhigh quality reproduction ofintense sounds necessitate aconsideration ofthemore exact equations ofwave motion ifdistortion due tothe generation ofextraneous frequencies intheairofthehorn itself istobe avoided. Similar considerations may beofsome importance incon- nection with the pick-up ofintense sounds. ‘The propagation ofwaves offinite displacement has interested physicists for more than acentury. In1808 Poisson derived an equation which shows that, ingeneral, asound wave cannot be propagated without achange inform and consequent generation of additional frequencies. This distortion iscaused bythenon-linearity ofair; that is,ifequal positive and negative increments ofpressure are impressed onamass ofairthechanges involume ofthemass willnot beequal; the volume change forthe positive pressure will beless than thevolume change fortheequal negative pressure. Anidea of the nature ofthe distortion can beobtained from the adiabatic curve AB forairasgiven inthefamiliar volume pressure indicator diagram (Fig. 1a). The undisturbed pressure and specific volume ofairare indicated bypoint PoV». Any deviation from the tangent through this point causes distortion and consequent generation ofextraneous frequencies. ‘The theoretical magnitudes ofthe waves ofextraneous frequencies are obtained from asolution ofthe exact differential equation ofwave propagation inair. The solution shows that the pressure ofthe second harmonic frequency, which isgenerated inthe air,increases with thefrequency andthemagnitude ofthefundamental*PublishedintheJanuary1988issueoftheJour.AconsSoe.Am [Eee Nente and A"Eethurs," Loua Speers and Mictophones," Bell Sy. Teak our S55 188. 189 g | 160 BELL SYSTEM TECHNICAL JOURNAL pressure and also with thedistance from thesound source. The solution also gives themagnitudes ofthewaves ofsum anddifference frequencies generated when two tones aresimultaneously impressed 20p'@ORO“COACCEEEC EE EEA AEE “COON eCeee“CEE ENE EEESCEPC CECCOCO ONE BEE NgeeSeya IN--CCCCSCCCCCCCOPRCEL ET(ee a SCEPC AAS SCCECCIPOCAS] SCECeCeSCE ECE SCC Ieee EEE x bcteh b ‘onthe air; these magnitudes also increase with distance from the source and with theproduct ofthefundamental pressures and, re- spectively, with thesumanddifference frequencies. EXTRANEOUS FREQUENCIES 161 THEORY oFPROPAGATION OFPLANE WavES oFFinrrg AMPLiTupE The derivation oftheexact differential equation forsound wave propagation inairinvolves thecontinuity equation, Newton's force equation and theequation expressing therelation between pressure and specific volume inagas. Since there may besome question asto theaccurate definition ofthedensity and force intheequation of motion asomewhat detailed discussion ofthissubject willbegiven. Following Rayleigh, letyand y+(ay/ax)dxbetheactualdistances attime ¢from theplane x=0toneighboring layers ofairwhose un- disturbed positions aredefined byxandx+dx,respectively, Fig. 1b. ~ vomit Po p Fig. 1b, ‘The displacement corresponding toyisthus £= y—« and the equation ofcontinuity ofthefluid is p=pa(dylax)* =pol+at/ax)—, 0) where pand poarethedensities ofthefluid inthedisturbed and un- disturbed states, respectively. Iftheeffect ofviscosity isneglected theexact equation ofmotion oftheelement ofmass p(dy/dx) -dsis PYgya8opie=—202 OPPast=ap“P=—ByGed or poatg/a)=—ap/ax, @ Pisthepressure atthepoint y(Fig. 1b)which moves with theair particle, notthepressure atafixed point. Except forvery large dis- placements these pressures arenearly thesame. From equations (1) and (2) #§/al =(dp[dp)-(1 +d¢/ax)-*-(a*¢/ax*). (3) *Lord Rayleigh, “Theory ofSound,” 2nd Ed., Vol. I,p.31. “aid 182 BELL SYSTEM TECHNICAL JOURNAL Byvirtue of,equation (1),equation (3)islinear in£only ifdp/dp =Ky-* ordp|do =—K,where =1/p=specific volume andKisa constant. This condition isnot satisfied during any ordinary varia- tions ofstate ofagas,butisapproximately satisfied when thevariations arevery small. Forisothermal changes wehave po=payandfor adiabatic changes: lbs =(oo/e)" =(olos)", @ where+istheratioofthespecificheatsandpsistheundisturbedatmospheric pressure. Ineither case, forvery small variations, the pvcurve ispractically identical with thetangent tothecurve, hence dp/dv ispractically constant (Fig. 1a). From equations (1),(3),(4)weobtain theexact equation ofadiabatic plane wave motion inanon-viscous fluid: aE/alt =aL+at/8x)--Mo*E/A2"), ) where @=ypo/po. This equation isgiven byRayleigh." *Rocard ¢ was first tocallattention tothegeneration ofharmonics intheair within anexponential horn. Histheoretical solution isbased ona plane wave equation inwhich theterm a¢/@# wasreplaced by oe 0b 8 (aESet ae(at) © Insupport ofthissubstitution Rocard cites Riemann’s §treatment of theproblem. However, Riemann’s analysis isbased ontheEulerian form ofthehydrodynamical equations whereas equation (5)isderived from theLagrangian equations. (For acomparison ofthese systems ofequations seeLamb) IntheLagrangian notation a¢/81 and S*e/at aretheexact values ofthevelocity andacceleration, respectively, Gftheparticle whose displacement from itsequilibrium position (x)is& Itistobenoted that inequation (2)theterm po,orundisturbed density, does notrepresent anapproximation. ‘Arigorous solution of(5)forthedisplacement §asanexplicit function of+and thasnotbeen obtained, Asafirstapproximation to equation (5)wetake FoHy 4nedS. Oy SLamb Donamical Theory ofSound" 2ndBayPIRshHePEEecstin snr oiPine i”ConteLafe FortplangungehenerLuftwelen vonenlicherSchwingung-eweitte” Going bhonabengen, Nov, 1860 Sin fifaredynamic, ethBd, Chapter 1 EXTRANEOUS FREQUENCIES 163 This approximation restricts thedilatation 9¢/2x tovalues small com- pared with unity ortheexcess pressure tovalues small compared with ‘vba, buttherestriction need notbeassevere aswould berequired for thelinear approximation: aE/att=catt/ax. Byamethod ofsuccessive approximations, carried tothe second approximation, Lamb*derivesthesolution of(7): §=acosw(t—x/c)+teeott=cos2u(t—x/e)](8) corresponding toamotion £=acoswtimposed ontheairatx=0, and assuming complete absence ofreflection. Byvirtue of(4)and (1) wehave forthepressure: b= poll —yak/ax +++). Neglecting theterms ofsmall amplitude, intheregion where 4x is large compared with thewave-length \wehave P= Pat bit Pn 0) where Pac=bo—vbo-((y +1)/8)-(#/A)a, bi==ybo-(wa/e) sinw(t—xe), br=ype((y+1)/4)-(w/e)-atxsin2a(t—x/c), or Di=MP, cos[w(t —x/c) +#/2], br=2P3 cos[2u(t —x/e) —7/2], (10) where Pi=ypowa/2hc, (ut) _ytl PE oxPIO abe “ Pi and Ps are thus the rms, fundamental and second harmonic pressures, respectively. Lamb®alsogivesasolutionof(7)forthecasewhentheforced motion atx=0is:£=£4Coswat+EyC08wal. Inaddition tothe two fundamentals and two second harmonics, the pressure now in- cludes components whose frequencies are, respectively, thesum and difference ofthetwo primary frequencies: bs=YP, cos[(wa +wa)(t —x/c) —x/2], ba=Py cos[(wa —o)(t —x[e) +4/2], : ’ 164 BELLSYSTEMTECHNICAL JOURNAL where \ “ , =Xt), PaPy (wa+on) P=TON aps . Med UHI PaPo (wa—n)xPeTO pe a ; and P,,Pyarethetwo r.m.s. fundamental pressures. Tfweextend Lamb's method ofsolution ofequation (7)tothethird approximation and again consider thecase £=@coswtatx=0we findthat intheregion kx>>1,ther.m.s. third harmonic pressure is 3(/y+1 wx)? =3 (TAL OF) 15raio(Gm e) os) Inthecase ofthegreatest r.m.s. fundamental pressure used inthe experiments, P;=8000barsat600c.p.s., equation (15)indicates that the third harmonic at400 cm from the source isabout 10db below the second harmonic. ‘Anapproximate correction fortheeffect ofattenuation inatube caused byviscosity andheat conduction canbeobtained byassuming that each oftheextraneous frequencies and the fundamental is attenuated asifitwere theonly wave present. Thus ther.m.s. value ofthefundamental atany point xisassumed tobe Pia Pet or dPyldx =—Ps, where P,isther.m.s. value ofthefundamental atthepoint x=0 and aithe measured attenuation factor for the fundamental. IfPeis ther.m.s. value ofthesecond harmonic atthepoint xandasis themeasured attenuation factor forthesecond harmonic intheabsence ofthefundamental, wehave byusing equation (12): aP,[dx =KP} —asP: =KPite*** —aaPs, (16) where arth 1ie.K=TON yp6 When ai=0and az=0,equation (16) isequivalent to(12). The solution of(16) which isconsistent with thefact that thesecond harmonic vanishes atx=0is Ps=(KP¢/(2ay —a2)[em—et], Hence hoya Psth Prey, PiaTa ap ay EXTRANEOUS FREQUENCIES 16s where ReOates 1 Forthetube used inthese experiments a2may betaken as24a; which gives R=1—ayx/28+--+, Similar correction factors were derived for the other extraneous fre- quencies measured intheexperiments. ‘MEASUREMENTS OF PLANE Waves oF FiniTE AMPLITUDE inATUBE The experimental work consisted ofmeasuring thesecond harmonic generated along atube. Measurements were also made ofthe sum and difference tones when two fundamental frequencies ofequal pressure were simultaneously impressed onthe airofthe tube. For thefundamental pressures and distances used intheexperiments, the magnitudes oftheother harmonics and higher order sum and difference frequencies were probably small. For high fundamental frequencies and pressures, however, these other tones are important, since they increase more rapidly with frequency and pressure than thesecond harmonic; forinstance, thethird harmonic pressure increases asthe square ofthe fundamental frequency and asthe third power ofthe fundamental pressure. Asinusoidal displacement, uniform over the cross section, was impressed ontheairatoneendofalong tube. The tube hadaninside diameter of3.8cmand was 1566 cmlong. Measurements were made inthefirst 705cmonly and theremainder ofthetube was used for obtaining anon-reflective termination. Asearch transmitter, comprising asmall tubeof0.08cminside diameter and 7.5cmlong, coupled toasmall condenser microphone,” was used for the measurements. Attenuation inthis search tube was sufficiently high toprevent either overloading themicrophone or . altering thesound wave propagated within thelong tube. The search transmitter was connected toastage ofamplification sooperated as topreclude non-linear distortion. This was followed byaband-pass filter which selected thefrequency desired inthemeasurements. The filter was terminated byameasuring circuit consisting ofahigh-gain amplifier and avacuum tube voltmeter. Adiagram ofthearrange- ment isshown inFig. 2. 1H1,C, Hargoon and P,B,Flanders, “An Efiiene Miniature Condenser Miro- phone Syaten Ba Sin Pek sensei St “aa 168 BELL SYSTEM TECHNICAL JOURNAL Toobtain reliable measurements throughout thelength ofthetest tube itwasnecessary toreduce standing waves toanegligible magni- tude. This wasaccomplished bylaying astrip offeltinthelast761cm ofthetube, terminating theendwith anacoustic resistance approxi- thately equal tothecharacteristic impedance ofthetube, andcarefully sealing upalljoints along thetube. The pressure variation inthe standing wavewas+0.3db,whichcorresponds toareflection coeffi- cient of0.035. nese wee ree eee a > | j(pes = j E.2extinari teansrTeR t(3) Fig.2-Apparatu focmearng extraneous Fequnces generate in caryintendstoundlwaves:Ryresistancesubstituteforreceiver. mine The oscillator current was supplied tothe loud speaker through a low-pass filter and themeasured harmonic content was found tobe 73dbbelow the fundamental. Pressure measurements inthe tube close totheloud speaker indicated that theharmonics generated inthe measuring circuit and loud speaker were more than 50dbbelow the fundamental pressure at2000 bars. ‘Acalibration ofthesearch transmitter was obtained bycomparison with asmall condenser microphone whose diaphragm was exposed directly tothesound wave atthesame position onthetesttube, see Fig.2.Thecalibrating microphone (C.T., Fig.2)hadbeen previously EXTRANEOUS FREQUENCIES 167 calibrated byathermophone.* The measuring circuit following the search transmitter was calibrated foreach frequency measured by introducing theoscillator current into thesearch transmitter circuit through theattenuator (a1,Fig. 2). The ratio ofthepressure ofthefrequency generated along the tube tothe fundamental pressure was measured bythe attenuator az, Fig. 2,atvarious holes along thetube inwhich thesearch transmitter was inserted. Ifanappreciable fraction oftheharmonic generated along the tube isreflected atthe end ofthe tube, the magnitude of thereflected component near thesource may becomparable with or larger than the harmonic generated between thesource and the point inquestion, This was found tobethe case when several measure- ments were made near thesource over adistance covering awave- length. The measured variation inthetotal pressure oftheharmonic over this distance was +t2.5 db whereas the variation for afunda- mental ofthis frequency, aspreviously stated, was +0.3db. There- fore themeasurements close tothereceiver may beinaccurate. Figure 3shows themeasured pressure ratio ofthegenerated second harmonic tothefundamental along thetube and thetheoretical curve === TnCE err (eee TTT PUD“eer A “00®uanceFRowSouRcEwwcentereas™ “°°MP calculated from equation (17). Each ofthethree experimental points plotted atabout 45cmfrom thereceiver istheaverage pressure ratio foraseries ofreadings taken over adistance ofawave-length along the tube. The measured and theoretical pressure ratios ofthesecond harmonic tothefundamental areshown asafunction offrequency and pressure inFigs. 4and 5. a 168 DELL SYSTEM TECHNICAL JOURNAL | “TTT 8, AT|SRa z -=—«_ DISTANCE FROM| ae| BCELoreea | i300 400-500 600 8001000 "2000 "3000 a a Fig.4Variation ofIndharmonic magnitude withfequeney offundamental T==eroe tT s per ||5-30 —_| aa) ft ;eeeeee omhes as Toes eo BES mcs soune'n wos Fig.$—Vatiaton of2ndharmonic pesure wth fundamental prec Figure 6shows themagnitude ofthefirstorder sumanddifference frequencies along thetube when twofrequencies ofequal pressure are simultaneously impressed ontheairinthetube. =| Fo a eeLeeeet PrTT) pee: [Ttaey et Ka af CT)Leer TTss1SWBE Fron cots Fig.6Maguitudeof summation anddiference fequenciesws.distancefromsure. EXTRANEOUS FREQUENCIES 1 ExponenTIAL Horn THEORY The second harmonic generated inany short section ofahorn is approximately thesame asthat generated inatube ofarea equal to the mean area ofthe section ofthehorn. Therefore, from the tube equation and the expression forthe change inpressure due tothe divergence ofthehorn themagnitude ofthegenerated second harmonic pressure atany point along thehorn can beobtained. ‘The r.m.s. value ofasmall excess pressure inanexponential horn ofsection S=See" isattenuated according tothelaw P=Pemt or dP/dx =—mP/2, where P,isther.m.s. pressure inthethroat ofthehorn (atx=0). The index oftaper misequal to4xf./c where f,isthecut-off frequency ofthe horn, From equation (12), therate atwhich ther.m.s. value ofthesecond harmonic increases along atube is dP; _yt1P? o_“de~TON"ypy'c ~RPE Ifitisassumed that thesame expression represents therate ofgenera- tion ofsecond harmonic along ahorn, and that both thefundamental and second harmonic diverge inthe same manner, the complete differential equation forP:becomes dP, m _ me “Fe~RPLPa=KPhe 7Pn where P;and P,, are, respectively, the r.m.s. fundamental pressures atthepoint inquestion (x)and inthethroat ofthehorn. The solution consistent with the condition P;=0atx=Ois Ps=(KP%,/(m/2) Lem? —em). (18) Since P;=P,,e-™!, theratio ofthesecond harmonic pressure tothe fundamental pressure atany point xinthehorn isthus pnKPw a Te apeet (19) +Accding t,t equation hepete ofthe cond homo rere oe echt Tern ye tn hance byMapas Eaton eats plain thepape “Lud Setarcnsoph icteric eee = where xf=(1—e™*)/(m/2). (20) Equations (19)and(20)indicate that thesecond harmonic atthe mouth ofanexponential horn oflength xisequivalent tothe‘second harmonic attheend ofatube oflength x’and ofuniform section, ‘equal totheareaofthethroat ofthehorn. ‘Thus thesecond harmonic inthemouth ofahorn having anindex oftaper m=0.075 cm™, cut- offfrequency 200c.p.s. andlength 78cmisequal tothesecond har- monic attheendofastraight tube 25cmlong andofthesame diameter asthe throat ofthe horn. EXPoNENTIAL HORN MEASUREMENTS: Measurements oftheoutput ofahorn attached toamoving coil receiver were made inanacoustically damped room. The horn had a throatdiameter of3.8cm,alengthof78cmandacut-off frequency of 200c.p.s. Thediaphragm oftheloud speaker wascoupled tothe throat ofthehornthrough astraight tube13cmlong.Afiltered single frequency tonewasimpressed ontheloudspeaker andthesound waspicked upbyasmall microphone infront ofthehorn. Thefunda- mental andsecond harmonic voltages from themicrophone amplifier were separated bymeans ofaband-pass filter andmeasured. The approximate acoustic power atthethroat ofthehorn was calculated from theknown efficiency oftheloud speaker and the electrical voltage and current supplied. The measured andcalculated ratios ofthesecond harmonic pressure tothefundamental pressure atthemouth ofthehorn, including the effects ofgeneration ofsecond harmonic inboth thehorn and the straight tube coupling thereceiver andthethroat ofthehorn, are shown inFig.7interms ofthesound output inwatts. Seeequations CO et eeeeee eae “oraresee 88 twat Pe! zo EXTRANEOUS FREQUENCIES im (17) and (19). The attenuation due toviscosity and heat loss inthe horn hasbeen neglected. ‘Anumber ofmeasurements atvarious microphone positions in front ofthehorn, shown bythedotted circles, indicate thedifficulty ofobtaining accurate results inaroom. ‘The average ofthemeasure- ments atanumber ofrandom positions infront ofthehorn atacon- stant sound power output and themeasurements atasingle position forvarious sound outputs gives the plotted curve which isprobably notgreatly different from that which would beobtained inopen air. Figure 8shows themeasured and calculated ratios ofthesecond harmonic pressure tothe fundamental pressure forvarious funda- mental frequencies. g.,,CO eaac.t_| |Peep oeroY TTT TT Se Fig. 82nd harmonic generated inanexponential horn s.fundamental frequen* Feneound output=10watts) equency DEMONSTRATION OFEXTRANEOUS FREQUENCIES Almost two hundred years ago Sorge, aGerman organist, and Tartini, anItalian violinist, discovered independently, apart from all theory, that the union oftwo loud independent tones produced a difference tone. That this isnotentirely asubjective tone produced bytheearbutisactually present intheairwas demonstrated by others some years later bytheuseofatuned resonator. With modern. apparatus consisting ofpower amplifiers, oscillators and tuned electro mechanical vibrators itisarelatively simple matter toshow notonly, the difference tone but the summation and harmonic tones aswell. ‘Anexponential horn was attached totheopen end ofthe1566 cm tube previously described but with thedamping material removed. ‘Amoving coil microphone placed infront ofthehorn picked upthe complex tone produced when two equal pure tones of600 and 940 cf 2 BELL SYSTEM TECHNICAL JOURNAL cps. were impressed ontheair. The microphone voltage was amplified andimpressed onsixtorsional vibrators tuned to340,600, 940, 1200, 1540 and 1880 c.p.s. Spherical mirrors attached tothe vibrators produced onascreen bands oflight theamplitudes ofwhich were approximately proportional totherelative pressures ofthe various frequencies inthecomplex tone. Forthehigher power inputs totheloud speaker used intheexperi- ment, thepresence ofthesum anddifference frequencies andthehar- monic frequencies waseasily observed. Atthese power outputs the quality ofthesound wasvery disagreeable andthefundamental tones could hardly bedistinguished. Coctuston ‘The theoretical andexperimental determinations oftheextraneous frequency waves generated intheairwithin atube areingood agree- ment asregards thevariation inmagnitude with frequency, distance from thesource andmagnitude oftheprimary tones. Themagnitude ofthesecond harmonic isvery nearly proportional tothedistance from thesource, tothefundamental frequency and tothesquare of theamplitude ofthefundamental pressure. When there aretwo primary tones, theextraneous frequencies generated intheairinclude, aswell astheharmonics oftheprimary tones, frequencies which are, respectively, thesum andthedifference oftheprimary frequencies andalsoother higher order tones. ‘Themagnitudes ofthesummation anddifference tones arevery nearly proportional tothedistance from thesource, totheproduct ofthemagnitudes ofthetwo primary pressures, andineach case, tothefrequency oftheparticular com- bination tone. Asregards theabsolute magnitudes ofthegenerated tones inthetube allofthemeasured values areabout 3dblower than the theoretical values. Good agreement wasobtained alsobetween theexperimental and theoretical determinations ofthesecond harmonic generated intheair within anexponential horn, asregards proportionality tothefunda- mental frequency andpower andalsoastoabsolute magnitude. In facttheagreement inabsolute magnitude wascloser forthehorn than forthetube, butnotmuch significance should beattached tothisfact, asthehorn theory isdeveloped from thetheoretical solution forthe tube and thehorn measurements areknown tobeless reliable than the tube measurements,